Micah Lea'alafa slotted a first-half penalty, while captain Henry Fa'arado did the same in the second half for the home side, who took advantage of sweltering conditions in Honiara to put in a better performance than they did last Friday.
New Zealand, who qualified for the 1982 and 2010 World Cup finals, now face the fifth-placed South American side -- currently twice champions Argentina -- in a two-legged playoff in November.
"It was a difficult day, we had a lot of changes and I was pleased just to get the job done," All Whites coach Anthony Hudson said.
"You have to give credit to the home team. They were giving everything and were playing for their pride and their country and they gave us a real test."
The All Whites had virtually assured themselves of the playoff following the win in Auckland and Hudson rested several key players including captain Chris Wood, who scored a hat-trick last Friday.
Despite an enterprising start, the home side's defensive frailties were again to the fore when they failed to clear a loose ball and Bevan poked home a 14th-minute rebound.
The visitors' second was also fortuitous when a Kosta Barbarouses cross deflected off Gagame's head and looped over goalkeeper Phillip Mango.
The home side put pressure on the New Zealand defence and goalkeeper Stefan Marinovic was forced into several good saves before Benjamin Totori was brought down in the box by Andrew Durante and Lea'alafa slotted home the spot kick.
While the hosts continued to take advantage of the New Zealanders' fatigue in temperatures around 30 degrees centigrade and in 80 percent humidity, they could only capitalise from the penalty spot with Fa'arodo converting in the 78th minute.
"It was a good result for us, because to go 2-0 down to a professional side and then come back it feels like a win," said the Solomons' Spanish coach Felipe Vega-Arango.
"They proved that they can compete. They never gave up. We had a lot of chances and their goalkeeper made a few good saves so it was a good score for us."
(Reporting by Greg Stutchbury in Wellington; Editing by Sudipto Ganguly)
